Knowledge of the curvature of surfaces is important in a number of applications such as flow simulations, computer graphics and animations, and pattern matching. It is of particular importance to applications dealing with evolving surface geometry. Such applications usually do not have smooth analytical forms for the surfaces forming the model geometry. Instead, they have to deal with discrete data consisting of points on the surface connected to form an unstructured mesh. Hence, it is important to be able to reliably estimate local curvatures at points on such discrete surfaces.
There are three classes of methods for estimating higher order information such as curvature from discrete surface information (e.g. nodes of a triangulation). The first class of methods builds a global or local parametrization for the discrete surface and computes curvature information from the derivatives of the parametrization. The second class of methods fits a smooth surface (such as a quadric) to a local set of points around the point of interest and uses the curvature of the surface at that point as the curvature estimate. These methods do not necessarily need the points to be connected to form a mesh in order to estimate the curvature, although initial estimation of the surface normal is facilitated by the availability of such a mesh. The smooth surface is usually chosen to be a limited form of a quadratic polynomials. The third class of methods estimates the curvature directly from the triangulation by a variety of methods such as discrete differential geometry.
In this work, the discrete differential geometry approach and quadric fitting approach were compared based on convergence studies. For general unstructured triangulations (no constraints on the structure or quality of the mesh), it was found that the discrete differential geometry did not exhibit convergence, the simple quadric fitting approach converged to the wrong solution while the extended quadric converged to the correct solution. Unfortunately, a vertex in a general unstructured mesh does not always have enough neighboring vertices to succesfully fit an extended quadric. To overcome this limitation, the extended quadric, extended patch method was devised in which the neighborhood of the vertex is extended by one level if the vertex does not have sufficient number of immediate neighbors. The method also includes enhancements using ghost vertices to estimate the curvature at surface mesh boundaries. The new method robustly estimates normals, principal curvatures, mean curvatures and Gaussian curvatures at vertices of general unstructured triangulations. The method has been tested on complex meshes and has proven itself to be accurate and reliable.
R. V. Garimella and B. K. Swartz, "Curvature Estimation for Unstructured Triangulations of Surfaces" , Los Alamos National Laboratory, 200